A trap or forecheck is not used to just get the puck as you simply suggest. The teams trapping do not want to play a puck possession game similar to the Hawks. They would rather you have the puck and they will force you into a bad read leading to a turnover leading to a scoring chance.

The different styles have as much to do with defense as it does offense. It is a philosophy. The Hawks want to have puck possession, any disruption (lost faceoffs) is a significant thing, as you can see they have a difficult time regaining possession.

A trap team or forecheck does not have that great of a reliance on possessing the puck, like the Hawks. They get a turnover and get a scoring chance, lose the puck? No big deal, they play a puck retrieval game. The Hawks do not.

You thinks it's a coincidence that the powerplay falters as soon as possession is lost?

Would you suggest a team with Kane size players try to emulate the Broad Street Bullies? Why would you think faceoffs (gaining puck possession) is not crucial to a puck possession team and load your roster with players poor at the dot.

It's a chess match out there gents, many ways to get a "W".

(agreed Bobby, was posting simultaneously)

Is that why the hawks lead the league in takeaways by a significant margin?

Faceoffs are definitely an issue, and not one that can be lightly disregarded. But this team has shown it can generally overcome it as it is so strong in many other areas.